## 38.16 Relatively pure modules

The notion of a module pure relative to a base was introduced in [GruRay].

Definition 38.16.1. Let $f : X \to S$ be a morphism of schemes which is of finite type. Let $\mathcal{F}$ be a finite type quasi-coherent $\mathcal{O}_ X$-module.

1. Let $s \in S$. We say $\mathcal{F}$ is pure along $X_ s$ if there is no impurity $(g : T \to S, t' \leadsto t, \xi )$ of $\mathcal{F}$ above $s$ with $(T, t) \to (S, s)$ an elementary étale neighbourhood.

2. We say $\mathcal{F}$ is universally pure along $X_ s$ if there does not exist any impurity of $\mathcal{F}$ above $s$.

3. We say that $X$ is pure along $X_ s$ if $\mathcal{O}_ X$ is pure along $X_ s$.

4. We say $\mathcal{F}$ is universally $S$-pure, or universally pure relative to $S$ if $\mathcal{F}$ is universally pure along $X_ s$ for every $s \in S$.

5. We say $\mathcal{F}$ is $S$-pure, or pure relative to $S$ if $\mathcal{F}$ is pure along $X_ s$ for every $s \in S$.

6. We say that $X$ is $S$-pure or pure relative to $S$ if $\mathcal{O}_ X$ is pure relative to $S$.

We intentionally restrict ourselves here to morphisms which are of finite type and not just morphisms which are locally of finite type, see Remark 38.16.2 for a discussion. In the situation of the definition Lemma 38.15.8 tells us that the following are equivalent

1. $\mathcal{F}$ is pure along $X_ s$,

2. there is no impurity $(g : T \to S, t' \leadsto t, \xi )$ with $g$ quasi-finite at $t$,

3. there does not exist any impurity of the form $(S^ h \to S, s' \leadsto s, \xi )$, where $S^ h$ is the henselization of $S$ at $s$.

If we denote $X^ h = X \times _ S S^ h$ and $\mathcal{F}^ h$ the pullback of $\mathcal{F}$ to $X^ h$, then we can formulate the last condition in the following more positive way:

1. All points of $\text{Ass}_{X^ h/S^ h}(\mathcal{F}^ h)$ specialize to points of $X_ s$.

In particular, it is clear that $\mathcal{F}$ is pure along $X_ s$ if and only if the pullback of $\mathcal{F}$ to $X \times _ S \mathop{\mathrm{Spec}}(\mathcal{O}_{S, s})$ is pure along $X_ s$.

Remark 38.16.2. Let $f : X \to S$ be a morphism which is locally of finite type and $\mathcal{F}$ a quasi-coherent finite type $\mathcal{O}_ X$-module. In this case it is still true that (1) and (2) above are equivalent because the proof of Lemma 38.15.5 does not use that $f$ is quasi-compact. It is also clear that (3) and (4) are equivalent. However, we don't know if (1) and (3) are equivalent. In this case it may sometimes be more convenient to define purity using the equivalent conditions (3) and (4) as is done in [GruRay]. On the other hand, for many applications it seems that the correct notion is really that of being universally pure.

A natural question to ask is if the property of being pure relative to the base is preserved by base change, i.e., if being pure is the same thing as being universally pure. It turns out that this is true over Noetherian base schemes (see Lemma 38.16.5), or if the sheaf is flat (see Lemmas 38.18.3 and 38.18.4). It is not true in general, even if the morphism and the sheaf are of finite presentation, see Examples, Section 109.39 for a counter example. First we match our usage of “universally” to the usual notion.

Lemma 38.16.3. Let $f : X \to S$ be a morphism of schemes which is of finite type. Let $\mathcal{F}$ be a finite type quasi-coherent $\mathcal{O}_ X$-module. Let $s \in S$. The following are equivalent

1. $\mathcal{F}$ is universally pure along $X_ s$, and

2. for every morphism of pointed schemes $(S', s') \to (S, s)$ the pullback $\mathcal{F}_{S'}$ is pure along $X_{s'}$.

In particular, $\mathcal{F}$ is universally pure relative to $S$ if and only if every base change $\mathcal{F}_{S'}$ of $\mathcal{F}$ is pure relative to $S'$.

Proof. This is formal. $\square$

Lemma 38.16.4. Let $f : X \to S$ be a morphism of schemes which is of finite type. Let $\mathcal{F}$ be a finite type quasi-coherent $\mathcal{O}_ X$-module. Let $s \in S$. Let $(S', s') \to (S, s)$ be a morphism of pointed schemes. If $S' \to S$ is quasi-finite at $s'$ and $\mathcal{F}$ is pure along $X_ s$, then $\mathcal{F}_{S'}$ is pure along $X_{s'}$.

Proof. It $(T \to S', t' \leadsto t, \xi )$ is an impurity of $\mathcal{F}_{S'}$ above $s'$ with $T \to S'$ quasi-finite at $t$, then $(T \to S, t' \to t, \xi )$ is an impurity of $\mathcal{F}$ above $s$ with $T \to S$ quasi-finite at $t$, see Morphisms, Lemma 29.20.12. Hence the lemma follows immediately from the characterization (2) of purity given following Definition 38.16.1. $\square$

Lemma 38.16.5. Let $f : X \to S$ be a morphism of schemes which is of finite type. Let $\mathcal{F}$ be a finite type quasi-coherent $\mathcal{O}_ X$-module. Let $s \in S$. If $\mathcal{O}_{S, s}$ is Noetherian then $\mathcal{F}$ is pure along $X_ s$ if and only if $\mathcal{F}$ is universally pure along $X_ s$.

Proof. First we may replace $S$ by $\mathop{\mathrm{Spec}}(\mathcal{O}_{S, s})$, i.e., we may assume that $S$ is Noetherian. Next, use Lemma 38.15.6 and characterization (2) of purity given in discussion following Definition 38.16.1 to conclude. $\square$

Purity satisfies flat descent.

Lemma 38.16.6. Let $f : X \to S$ be a morphism of schemes which is of finite type. Let $\mathcal{F}$ be a finite type quasi-coherent $\mathcal{O}_ X$-module. Let $s \in S$. Let $(S', s') \to (S, s)$ be a morphism of pointed schemes. Assume $S' \to S$ is flat at $s'$.

1. If $\mathcal{F}_{S'}$ is pure along $X_{s'}$, then $\mathcal{F}$ is pure along $X_ s$.

2. If $\mathcal{F}_{S'}$ is universally pure along $X_{s'}$, then $\mathcal{F}$ is universally pure along $X_ s$.

Proof. Let $(T \to S, t' \leadsto t, \xi )$ be an impurity of $\mathcal{F}$ above $s$. Set $T_1 = T \times _ S S'$, and let $t_1$ be the unique point of $T_1$ mapping to $t$ and $s'$. Since $T_1 \to T$ is flat at $t_1$, see Morphisms, Lemma 29.25.8, there exists a specialization $t'_1 \leadsto t_1$ lying over $t' \leadsto t$, see Algebra, Section 10.41. Choose a point $\xi _1 \in X_{t'_1}$ which corresponds to a generic point of $\mathop{\mathrm{Spec}}(\kappa (t'_1) \otimes _{\kappa (t')} \kappa (\xi ))$, see Schemes, Lemma 26.17.5. By Divisors, Lemma 31.7.3 we see that $\xi _1 \in \text{Ass}_{X_{T_1}/T_1}(\mathcal{F}_{T_1})$. As the Zariski closure of $\{ \xi _1\}$ in $X_{T_1}$ maps into the Zariski closure of $\{ \xi \}$ in $X_ T$ we conclude that this closure is disjoint from $X_{t_1}$. Hence $(T_1 \to S', t'_1 \leadsto t_1, \xi _1)$ is an impurity of $\mathcal{F}_{S'}$ above $s'$. In other words we have proved the contrapositive to part (2) of the lemma. Finally, if $(T, t) \to (S, s)$ is an elementary étale neighbourhood, then $(T_1, t_1) \to (S', s')$ is an elementary étale neighbourhood too, and in this way we see that (1) holds. $\square$

Lemma 38.16.7. Let $i : Z \to X$ be a closed immersion of schemes of finite type over a scheme $S$. Let $s \in S$. Let $\mathcal{F}$ be a finite type, quasi-coherent sheaf on $Z$. Then $\mathcal{F}$ is (universally) pure along $Z_ s$ if and only if $i_*\mathcal{F}$ is (universally) pure along $X_ s$.

Proof. This follows from Divisors, Lemma 31.8.3. $\square$

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